Poisson distribution

Definition: $X\sim Po(\lambda )$

• Discrete random variable
• With assumptions:
  • Events occur singly, randomly and independently
  • at consistent rate in a given interval of space/time
• No fixed range of value, extreme values can be happened, but the chance is insignificant
• $\lambda$ is both mean and variance
• Example: number of misprints on a page

Probability mass function

• $P(X=i)=e^{-\lambda }\frac{{\lambda }^i}{i!}$ for $i=0,1,2,...,\infty$
  • $\sum _{i=0}^{\infty }p(i)=e^{-\lambda }\sum _{i=0}^{\infty }\frac{{\lambda }^i}{i!}=e^{-\lambda }.e^{\lambda }=1$

Expected value $E[X]=\lambda$

Variance $Var(X)=\lambda$

Poisson approximation for binomial distribution:

• Can be used to approximate binomial distribution when:
  • $n$ is large
  • $p$ is small enough so that $\lambda =np$ is of moderate size

Computing the poisson distribution function:

• $\frac{P(X=i+1)}{P(X=i)}=\frac{\lambda }{i+1}$

Sum of independent Poisson distribution:

• $X\sim Po({\lambda }_1)$ and $Y\sim Po({\lambda }_2)$, then $X+Y\sim Po({\lambda }_1+{\lambda }_2)$